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<p>a ring  with .  Equivalently, it is both a <a href="page.php?w=subgroup">subgroup</a> of  and a <a href="page.php?w=submonoid">submonoid</a> of .</p>

<p>Equivalently,  is a subring <a href="page.php?w=if_and_only_if">if and only if</a> it contains the multiplicative identity of , and is <a href="page.php?w=Closure_%28mathematics%29">closed</a> under multiplication and subtraction. This is sometimes known as the subring test.</p>

<p><big> Variations </big></p>
<p>Some mathematicians define rings without requiring the existence of a multiplicative identity</p><p>
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